B. com 5th- Sem

A) DEFINITION

A matrix is defined as a rectangular arrangement of numbers in rows and columns. e.g.,

NOTE:

1) A matrix is always denoted by a capital letters, like A, B, C, ...etc.

2) The number which are listed within brackets are known as

" Elements or Entries or Members" of the matrix.

3) Generally [ ] or ( ) brackets are used to donate a matrix.

4) The horizontal lines are known as " ROWS " and vertical lines are known as " COLUMNS ". Thus the matrix A has two rows and three columns. We say to be a rectangular (since, the number of rows is different from number of columns) matrix of order 2x3 and B to be a square matrix or order 3x3 (since, the number of rows is same as the number of columns).

• In other words we can say A matrix having m rows and n columns is called a matrix of order m x n (read as m by n).

REAL METRIX : if the elements of a matrix be all real then the matrix is called a real matrix.

B) SOME DEFINITION OF METRIX:

a) ROW MATRIX: A matrix whose elements are arranged in one row only is called a row matrix.

For examples, [a₁ a₂ a₃ ........aₙ] is a row matrix. Clearly, the order of the matrix is 1 x n.

b) COLUMN MATRIX: A matrix whose elements are arranged in one Column only is called a Column matrix. For example,

b₁

b₂

b₃

bᵤ is a column matrix. Clearly, the order of the matrix is m x 1.

c) NULL or ZERO MATRIX : A matrix whose every element is zero is called a null matrix or zero matrix. A null matrix is generally denoted by O. A null matrix of order u x n is denoted by

d) SQUARE MATRIX: A matrix whose number of rows and columns are equal is called a square matrix. If number of rows = number of columns= n, then the matrix is called a square matrix of order n (or n x n square matrix). For example, a₁ b₁ c₁

a₂ b₂ c₂

a₃ b₃ c₃

is a square matrix of order 3 (or third order matrix).

DIAGONAL ELEMENTS: If A =[aᵢⱼ] be a square matrix, then elements where I = j are called diagonal elements and the straight line on which they lie is called the Principal diagonal or simply diagonal of the square matrix. For example, for the square matrix

1 3 5

4 6 8

2 7 4

is the diagonal elements are 1, 6, 4.

d) DIAGONAL MATRIX: A square matrix whose all the elements Except the elements in principal diagonal are zero is called a diagonal matrix. For example

a₁ 0 0

0 b₂ 0

0 0 c₃

is a diagonal matrix of order 3. And principal diagonal is a₁ b₂ c₃

2 0

0 3

is a diagonal matrix of order 2. And principal diagonal is 2 3.

NOTE:

1) A square matrix has two diagonals. The diagonal extending from left-hand top corner to right hand bottom corner is known as PRINCIPAL or MAIN or LEADING diagonal.

2) Atleast one element of the principal diagonal must be non-zero.

e) SCALAR MATRIX: A diagonal matrix whose elements on the principal diagonal are non-zero and all equal to each others is called scalar matrix for example

1) -2 0 2) 7 0 0

0 -2 0 7 0

0 0 7

f) UNIT or IDENTITY MATRIX: A scalar matrix whose elements on the principal diagonal are all 1 is called a unit or identify matrix. An unit matrix is generally denoted by I. Sometimes the order of the unit matrix is also written as suffix of I. For example. I₂ and I₃ are unit metrices of order 2 and 3 where

I₂ = 1 0 and I₃= 1 0 0

0 1 0 1 0

0 0 1

NOTE:

1) All the diagonal elements of the unit matrix should be + 1 only.

2) The diagonal matrix, scalar matrix and unit matrix all the square matrices i.e., in each of them the number of rows= the number of columns.

g) UPPER TRIANGULAR MATRIX:

This is a square matrix whose each element below the princpal diagonal is zero. Example

1) 2 3 2) 2 4 0 3) 2 0 0

0 1 0 3 1 0 0 1

0 0 5 0 0 5

h) LOWER TRIANGULAR MATRIX:

This is a square matrix whose elements above the principle diagonal is zero. Example.

1) 2 0 2) 2 0 0 3) 0 0 0

1 3 3 1 0 2 0 0

4 0 1 3 1 0

****************"*"*********""""***********

* ADDITION OF TWO MATRICES *

Two matrices can be added or subtracted only if they are of the same order and this is done by adding or subtracting the corresponding elements of two matrices.

PROPERTIES OF MATRIX ADDITION:

1) A+ B= B + A

2) A +(B+C)= (A+B)+C

3) A + O= O +A = A

Where A, B, C are the matrices of the same order and O is the null matrix of the same order.

Exercise - A

1) If A= 2 5 & B= 1 -3

-3 7 2 5

Find A+ B. 3 -2

-1 12

2) If A= 3 -6 & B= 1 -3

9 3 3 0

Find 2A+ B. 7 -15

21 6

3) If A= 2 -3 & B= -1 6

4 5 3 2

Find 2A+ 5B. -1 24

23 20

4) If M= 2 0 & N= 2 0

1 2 -1 2

Find M+ 2N. 6 0

-1 6

5) If A= 2 0 and B= 1 2

-3 1 3 1

Find 3A+ 4B. 10 8

3 7

6) If A= 1 4 and B= -4 -1

2 3 -3 -2

Find 2A+ B. -2 7

1 4

7) If A= -3 2 1 & B= 3 5 1

1 -4 7 -1 4 -2

Find A+ B. 0 7 2

0 0 5

8) If A= diag(1 -1 2) and B= diag(2 3 -1), find

A) A+ B. diag (3 2 1)

B) 3A + 4B. diag(11 9 2)

Exercise - B

1) If A= 3 -1 7 & B= 2 -2 -1

0 1 2 1 2 3

Find A+ B. 5 -3 6

1 3 5

2) If A= 0 2 3 & B= 7 6 3

2 1 4 1 4 5

Find 2A+ 3B. 21 22 15

7 14 23

3) if A = 1 2 3 & B= 4 -2 -2

0 3 - 5 6 2 -1

1 0 0 1 -2 3

find 2A + 3B. 14 -2 0

18 12 -13

5 -6 9

4) 1 2 3 3 -1 3

A=2 3 1 & B = -1 0 2

Find 2A + 3B. 5 1 15

1 6 5

5) A= 2 3 4 & B= 3 0 5

0 4 6 5 3 2

5 8 9 0 4 7

find 3A - 2B. 0 9 2

-10 6 14

15 16 13

6) A=1 -3 2 B= 2 -1 -1 C = -3 4 -1

2 0 2 1 0 -1 -3 0 -1

Find 2A + 3B+ C

7) If A=1 3 5 and B= 4 2 3

0 4 1 1 3 -7

Verify 2(A+B)= 2A +2B

* SUBSTRACTION OF MATRICES *

Exercise - C

1) If A = 2 4 and B = 3 6

5 6 5 4

Find A- B. -1 -2

0 2

2) If A= 3 -6 & B = 1 -3

9 3 3 0

Find A- 2B. 1 0

3 4

3) If M= 2 0 and N= 2 0

1 2 -1 2

Find:

A) 2M - 3N. -2 0

5 -2

B) 2(M - 3N). 16 0

8 -8

C) 2M - N. 2 0

3 2

D) 7M - 5N. 4 0

12 4

4) If A= 7 6 3 and B = 0 2 3

1 4 5 2 1 4

Find 3B - A. 21 16 6

1 11 11

5) If A= diag(2 -5 9), B= diag(1 1 -4) and C=(-6 3 4) find

i) A- 2B. diag(0 -7 17)

ii) B+C - 2A. diag(-9 14 -18)

iii) 2A + 3B-5C. diag(37 -22 -14)

Exercise - D

1) If A = 2 4 and B = 3 6

5 6 5 4

Find X if 3A + 4B - 2X= 0. 9 18

35/2 22

2) If A= 5 -4 & B= 3 -2

-11 6 -1 4 find the Matrix X when A+ 2X = 3B. 2 -1

4 3

3) If B= -2 2 0 C= 2 0 -2

3 1 4 7 1 6 find the value of A as the relation 2A- 3B + 5C = 0. -8 3 5

-13 -1 -9

4) If A= diag(2 -5 9), B= diag (1 1 -4) and C= diag (-6 3 4) find

A) A- 2B. diag(0 -7 17)

B) B+ C - 2A. diag(-9 14 -18)

C) 2A + 3B - 5C. diag(37 -22 -14)

5) Let A= -1 0 2 B=0 -2 5 & C= 1 -5 2

3 1 4 1 -3 1 6 0 -4 Compute 2A - 3B + 4C. 2 -14 -3

27 11 -11

** EQUALITY OF TWO MATRICES:

Two matrices are said to be equal if

1) they are of same orders.

2) each element of one is equal to the corresponding element of the other. For example, if A and B are matrices both of order 2x 3 and

EXERCISE-E

1) If A= 2a - b 4 = 2 4

3 a+2b 3 6 find the value of a and b. 2,2

2) If A=2 a & B = -2 3 & C = c 9

-3 5 7 b -1 -11

and 5A + 2B = C find a, b, c. 6/5, -18,6

3) If A= x y B= 1 -1 & C= 3 5

z t 0 2 4 6 with the relation 2A+ 3B = 5C then find the value of x,y,z and t. 6,14,10,12

4) If A= 2 0 -4 B= 1 -2 0 C= 2

1 4 2 2 -1 3 0

1 Find the Matrix C with the relation (3A - 2B)C = D.

05) A= 1 2 3 B= 1 0 2 C= 4 4 10

-1 -3 2 3 4 5 4 2 14 find the value of k if 2A+ kB=C. 2

Type- 2

1) If X + Y = 7 0 and X - Y= 4 -1

2 3 5 -2

2) A+ B= 2 2 & 2A+ 3B= 5 4

0 2 0 5 then find the Matrix A and B.

3) If A+B= 1 2 0 & A - B= 3 0 2

3 5 4 1 1 0

Find the metrices A and B

2 1 1 -1 1 -1

2 3 2 1 2 2

4) If A+B= 2 2 and A - B= 5 4

0 2 0 5

Find the metrices A and B.

7/2 3 -3/2 -1

0 7 0 -3/2

5) If 2P + Q= 4 5 & P + 2Q= 2 4

3 8 3 1 Then find the value of P+ Q. 2 3

2 3

6) If 2A+ B= 4 4 7 & A- 2B = -3 2 1

7 3 4 1 -1 2 find the Matrix A and B

1 2 3 2 0 1

3 1 2 1 1 0

7) If 2A+B= 4 7 16

7 -3 12

13 6 2

And 3B - A= 5 0 13

7 -2 8

11 4 -1

Find metrices A and B

2 1 6 1 3 5

3 -1 4 2 -1 4

5 2 0 4 2 1

8) If A + 2B = 1 2 0

6 -3 3

-5 3 1

and 2A - B= 2 -1 5

2 -1 6

0 1 2

Find metrices A and B.

0 5/3 -5/3 1 1/3 10/4

10/3 -5/3 0 -2/3 1/3 3

-10/3 5/3 0 5/4 -1/3 1

9) x - y = 1 1 0 and x+ y= 3 5 1

1 1 0 -1 1 4

1 0 0 11 8 9 find Matrix x and y. 2 3 1 1 2 0

0 1 2 -1 0 2

6 4 0 5 4 0

MULTIPLICATION OF A MATRIX BY A SCALAR :

Multiplying a matrix by a real number is called multiplication of a matrix by a scalar.

If k is a real number (scalar) and A be a matrix, then their product kA is defined by the matrix of the same order as A, whose elements at each position is k times that of A. i.e., if A = [aᵢⱼ)ᵤₓᵥ, thenkA =ln([kij]ᵤₓᵥ

PROPERTIES OF MULTIPLICATION OF A MATRIX BY A SCALAR:

Let p,q are any two scalar and A and B are metrices of the same order then

PROP 1). (p+q)A= pA + qA

PROP 2) p(A+B)= pA + pB.

PROP 3) p(qA) = (p q)A or

(p q)A = p(qA)

Exercise - F

1) If A = 1 2 B= 3 4 C= -1 0

5 -4 0 2 2 -2

find i) ABC

ii) (A+B)C

iii) A² - BC

iv) AC + B²

v) (A+ B) (A - B)

vi) AC + B²

2) If A= 3 5 & B= 1 6

1 -2 -4 3 show AB≠ BA

3) A= 2 0 & B= 3 0

3 1 3 2 show AB = BA

4) A= 2 1 & B= 1 -2

1 2 -1 2 show AB = B

5) If A = 1 2 3 and B = 6 -2 -3

1 3 3 -1 1 0

-1 0 1 1 2 4

Evaluate AB , BA

6) If A = 1 -2 1

-1 2 -1 show A² = A

-2 4 -2

7) If A= 1 -1 and B= 1 1

-1 1 1 1

prove AB=0.

8) If A = 1 2 1 & B= 1 4 0

1 -1 1 -1 2 2

2 3 1 0 0 2

Find AB - 2B

10) If A= 0 1 2 B = 2 1 3

1 2 3 -1 0 1

3 1 1 3 -1 4

Verify AB≠ BA

11) If A = 1 2 3 and B= 6 -2 -3

1 3 3 -1 1 0

1 2 4 -1 0 1

Check AB = BA

12) If A= 1 & B= 1 2 3 find BA

15) If A is 3 -2 0 & B= 2 find BA

16) If A= 2 -1

- 1 2 show that A²- 4A+ 3I = 0, where I is 2 x2 unit Matrix and 0 is 2x2 zero Matrix.

17) If I= 1 0 & B= 3 2

0 1 2 1 show that A²- 4A - I= 0, where 0 is the zero Matrix of order 2

18) If A= 4 3 and B= 1 0

2 5 0 1 find the values of x and y so that A² - xA + yI =0, where 0 is the zero Matrix of order 2. 9, 14

19) If A = 2 -1 show A² - 4A + 3I =0

-1 2

20) If A = 2 0 1 find A² - 5A + 6I

2 1 3

1 -1 0

MATRIX NOTATION:

Exercise - G

1) construct a 2 x 2 matrix A = [aᵢⱼ] whose elements are given by

A) aᵢⱼ = (i+j)²/2. 2 9/2

9/2 8

B) aᵢⱼ = (i -j)²/2. 0 1/2

1/2 0

C) aᵢⱼ= (i- 2j)²/2. 1/2 9/2

0 2

D) aᵢⱼ= (2i-j)/2. 1/2 0

3/2 1

F) aᵢⱼ= |-3i+ j|/2. 1 1/2

5/2 2

2) Construct a 2x3 Matrix whose elements aᵢⱼ are given by:

A) aᵢⱼ= i. j. 1 2 3

2 4 6

B) aᵢⱼ= 2i - j. 1 0 -2

3 2 1

3) Construct a 2x3 matrix A= [aᵢⱼ] whose elements are given by

A) aᵢⱼ= (i - j)/(i+j). 0 -1/3 -1/2

1/3 0 -1/5

B) aᵢⱼ= i. j. 1 2 3

2 4 6

C) aᵢⱼ = 2i - j. 1 0 -1

3 2 1

Exercise - H

1)If A = 2 4 -1 and B = 3 4

-1 0 2 -1 2

2 1

Find

i) A′

ii) A′ +B′

iii) (A - B)ᵗ

iv) (AB)′

v) Aᵗ Bᵗ

2) If A= 2 1 and B= 1 -2

3 4 -1 1

Prove (AB)'= B ' A '

3) If A = 0 -1

2 3

Prove (A ')' = A

4) If A = 4 2 -1 and B= 2 3

3 -7 2 -3 0

1 5

Find possible or not A+B, A - B , AB, BA

Exercise - I

** A square Matrix A is a symmetric Matrix iff A'= A

** A square Matrix A is a skew-symmetric Matrix iff A'= - A

*** Sum of a symmetric and skew-symmetric Matrix= 1/2 (A+ A')+ 1/2 (A - A')

1) If A= 3 -1 1

-1 2 5

1 5 -2 is a symmetric Matrix.

2) if A= 0 2 -3

-2 0 5

3 -5 0 is a skew-symmetric Matrix.

3) If A= 2 3

4 5 Prove A- A' is a skew-symmetric Matrix

4) if A= 3 -4

1 -1 show that A - A' is a skew-symmetric Matrix.

5) If A= 5 2 x

y z -3

4 t -7 is a symmetric Matrix, find x,y z, t. 4, 2 , -3

6) Express the Matrix A as the sum of symmetric and skew-symmetric Matrix

A= 4 2 -1

3 5 7

1 -2 1

7) Express the Matrix A as the sum of symmetric and skew-symmetric Matrix

A= 3 -4

1 -1

8) Let A= 3 2 7

1 4 3

2 5 7 Find matrix X and Y such that X+Y = A, where X is a symmetric and Y is a skew-symmetric matrix.

9) If A= 2 4

5 6 Prove A+A' is a symmetric matrix where A' is the transpose of A.

EXERCISE -J

Find the determinants of

1) 2 -3

4 1 14

2) 4 2 -3

1 4 4 4

3) 2 3 -1

4 1 2

3 1 -1 23

4) 1 0 1

2 -1 1

3 0 1

5) 2 3 -4

1 3 -1

3 1b 4 37

***Evaluate Expanding by 2nd row:

6) 2 0 -1

1 3 2

-2 4 1 -20

2) 3 -2 1

1 0 1

2 4 5 2

EXERCISE - K

A) Find the cofactors of a₂₁ & a₂₂

1) A= 2 -3

1 4 3, 2

2) -4 1

3 -2 -1,-4

B) Find the cofactors of a₁₃, a₂₂, a₃₂

1) 0 -1 2

A= -3 4 -5

6 -7 8 -3,-12,-6

2) 2 4 -6

A= -8 4b 3

2 -4 1 24, 14, 42

EXERCISE - L

Find the adjoint of the following:

1) a b d -b

c d -c a

2) -3 5 4 -5

2 4 -2 -3

3) p q s -q

r s -r p

4) - 2 3 4 -3

- 5 4 5 -2

5) 1 2 2 -3 2 2

2 1 2 2 -3 2

2 2 1 2 2 -3

6) 1 2 5 2 3 -13

2 3 1 -3 6 9

-1 1 1 5 -3 -1

9) -4 -3 -3

If A= 1 0 1

4 4 3

Show that adj A= A

10) 1 -2 3

If A= 0 2 -1

-4 5 2

Find A(adj). 25 0 0

0 25 0

0 0 25

EXERCISE - M

State which of the following are INVERTIBLE :

1) 2 -3

2 3 YES

2) 4 -1

-4 1. NO

3) 5 2 -3

4 -5 2 NO

0 3 -2

4) 1 1 -1

2 3 1. YES.

1 -1 -2

*** FIND the condition for which the following matrices are INVERTIBLE

1) a b

c d ad-bc≠ 0

Find the inverse of following:

1) 2 -1 4/11 1/11

3 4 -3/11 2/11

2) 0 1 0 1

1 0 1 0

5) a b (1+bc)/a -b

c (1+bc)/a - c a

6) 2 5 1/17 -5/17

-3 1 3/17 2/17

7) If A=3 2 & B= 6 7

7 5 8 9 then find the value of (AB)⁻¹. -47 39/2

41 -17

9) 1 3 3 7 -3 -3

1 4 3 -1 1 0

1 3 4 -1 0 1

10) 1 2 3 -5/18 1/18 7/18

2 3 1 1/18 7/18 -5/18

3 1 2 7/18 -5/18 1/18

11) 1 2 5 4/27 17/27 3/27

1 -1 -1 -1/27 11/27 2/9

2 3 -1 5/27 1/27 -1/9

12) 2 0 -1 3 -1 1

5 1 0 -15 6 -5

0 1 3 5 -2 2

*** FIND A If:

1) 2 - 1 2 1

A⁻¹= -3 2 3 2

2) 2 -1 3 -2 2 4

A⁻¹=1 1 1 0 1 -1

1 -1 1 2 -1 -3

EXERCISE -N

Prove that

1) If A = 1 1

2 3

prove A² - 4A +5I=0. Hence find A⁻¹.

3/5 1/5

-2/5 1/5

2) If A = 4 5 satisfies A² - I=10A

5 6 Hence find A⁻¹

3) A= 2 -3

3 4 satisfies the equation x²- 6x+17= 0. Hence find A⁻¹. 4/17 3/17

-3/17 2/17

4) If A= 3 2

2 1

Prove A² - 4A - I= 0 where I= 2x2 metrics and 0 is Null metrics. Then find inverse of A.

5) If A= 2 -1

-1 2

Prove A² - 4A + 3I = 0

6) A= 2 3

1 2 verify A²- 4A + I= 0, then find A⁻¹. 2 -3

-1 2

7) If A= 4 5

2 1 find A⁻¹ and show that 2A⁻¹ =9I - A.

8) IfA= 2 2 0 then A³-13A+12I=0

2 1 1 . Hence find A⁻¹

-7 2 -3

9) A= 1 2 3

3 -2 1 show A³ -23A-40I=0

4 2 1 Hence find A⁻¹

10) If A= 1 0 -2

-2 -1 2

3 4 1 Show that A³- A²- 3A - I = O. Hence find A⁻¹. -9 -8 -2

8 7 2

-5 -4 -1

EXERCISE -P

Solve:

A) 5x - 7y =2, 7x - 5y =3,. 11/24, 1/24

B) x - 2y -4 =0, -3x +5y+7 =0, -6,-5

C) 3x+ 4y =5, x-y =-3. -1,2

D) ax+ by=c, a²x +b²y =c².

E) 3/x - 5/y =1 , 2/x +3/y = 7.

F) a/x -b/y =a , a/y - b/x =b.

G) 5x +7y =-2, 4x +6y+3=0. 9/2,-7/2

H) 5x +2y =3, 3x +2y=5. -1,4

I) 3x +7y =4, x +2y+1=0. -15,7

A) x+y-z=3, 2x+3y+z=5 , 3x-y-7z=1. 3,1,1

B) x+ 2y+z=7; x+ 3z=11; 2x-3y=1. 2,1,3

C) 2y-3z=0, x+3y= -4, 3x+4y =3

E) x+y-6z=0, -3x+y+2z=0, x-y+2z=0

F) x+y+z=4, 2x-y+2z=5 , x-2y-z=-3

F) (a+b)x - (a-b)y= 4ab,

(a-b)x + (a+b)y = 2(a² -b²)

G) 1/x +2/y+ 3/z=2

2/x +4/y +5/z =3

3/x +5/y+6/z= 4

H) 2/x +3/y +2=0

5/y - 2/z -4 =0

3/z +4/x +7=0

Continue.......

MATHS FOR ALL

B. com 5th- Sem

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